IS14277:1996 ( Reaffirmed 2002 ) STATISTICAL INTERPRETATION OF TEST RESULTS - ESTIMATION OF MEAN, STANDARD DEVIATION AND REGRESSION COEFFICIENTCONFIDENCE INTERVAL ICS 03.120.30 0 BIS 1996 BUREAU MANAK OF INDIAN STANDARDS BHAVAN, 9 BAHADUR SHAH ZAFAR MARG NEW DELHI 110002 Price Group August 1996 3 Statistical Methods for Quality and Reliability Sectional Committee, MSD 3 FOREWORD This Indian Standard was adopted by the Bureau of Indian Standards, after the draft finalized by the Statistical Methods for Quality and Reliability Sectional Committee had been approved by the Management and Systems Division Council. Industrial experimentation is often performed to estimate some unknownvalues. Thesevalues are usually parameters (constants, such as, mean thickness of metal sheets; variance of tensile strength of wire; relationship between carbon content and tensile strength of steel, etc) of a probability distribution or function of these parameters. The present standard deals with estimation of parameters of a normal population on the basis of series of test results on items in a sample drawn from this population, and discusses point estimates and interval estimates for mean, standard deviation and regression coefficient. It is not concerned with the calculation of an interval containing, with a fixed probability, at least a given percentage of the population (that is, statistical tolerance interval) which is covered in IS 13131 : 1991 `Statistical tolerance interval - Methods for determination'. A point estimate is a single value which is used to estimate the parameter in question. A point estimate is often inadequate as an estimate of a parameter since it rarely coincides with thevalue of the parameter and does not indicate how far away this estimate is from the true value of the parameter. One way of expressing this uncertainty is to specify an interval as an estimate instead of a single value, stating that the interval thus calculated includes the true value of the population parameter, has a specified high probability. Such an interval is called a confidence interval. The confidence interval is obtained as a function of the test results on items in the sample of observations and therefore is a random interval. Associated with it is a confidence level (sometimes termed as confidence coefficient), which is the probability, usually expressed as a percentage, that the interval does contain the parameter of the population. Ifwe choose an interval such that the probability that it contains the value of the population parameter is 1 - a, then we say that the interval is a 100 (1 - a ) percent confidence interval for the parameter and 1 - a is known as the confidence level. Generally 9.5percent and 99 percent confidence intervals are constructed. An informative list of published Indian Standards on Statistical Methods is given at Annex A. The committee responsible for the formulation of this standard is given at Annex B. IS 14277 : 1996 Indian Standard STATISTICALINTERPRETATIONOF TESTRESULTS-ESTIMATIONOFMEAN, STANDARDDEVIATIONAND REGRESSIONCOEFFICIENTCONFIDENCEINTERVAL 1 SCOPE S= This standard specifies the statistical treatment of test results needed to calculate confidence interval for the mean, standard deviation and regression coefficient of a population. 2 REFERENCES The following Indian Standards are necessary adjuncts to thiS standard: IS No. 6200 Title II 1 n_l i i 1 (Xi - q2 or .S= J 1 n-l 5 [ i-1 x? - I 2 HXi `=; I where Xi = the value of the ith test result (i = 1,2, .... n) ; (Part 1) : 1995 6200 (Part 2) : 1977 7920 (Part 1) : 1994 9300 (Part 2) : 19S9 3 DEFINITIONS Statistical tests of'significance: Part 1 t-, Normal and F tests (second revision) Statistical tests of significance: Part 2 x2 test (first revision) Statistical vocabulary and symbols : Part 1 Probability and general statistical terms Statistical models for industrial applications: Part 2 Continuous models (first revi.sion) n = the total number of test results; and x' = the sample mean calculated as in 4.1. 4.3 Estimation of the Regression Coeffkient Consider the linear regression of an independent variable (X) on the dependent variable (Y), E(YjX=x) =/To +/%x Given n( >2) pairs of test results (xi, yi), i = 1,2,..., n, the estimate of& is bl = $ sx For the purpose of this standard, the definitions given in IS 7920 (Part 1) : 1994 shall apply. 4 POINT ESTIMATION 4.1 Estimation of the Mean A point estimate of the population mean (designated as 1~)is given by the sample mean (3 and is obtained as follows: - =- 1 " x Z Xi ni,l where sx is the estimate of standard deviation ofX and s, is the estimate of covariance ofXand Y. sx is given by the expression in 4.2 and sq is given by the following expression: 1 (Xi -q(Yi -2 sv = x.i I 1 or s, =~ [CXiyi - ~~ 5 CONFIDENCE INTERVAL FOR THE MEAN 5.1 Standard Deviation Known The confidence interval for the population mean is calculated from the point estimate @ of the mean and the known standard deviation ( a). A 100 (1 - a) percent confidence interval for the population mean of a normally distributed variable with meanp and variance u is given by: 1 wherexl, sample. x2, . . . . xn are test results of n items in the 4.2 Estimation of the Standard Deviation The point estimate of the population standard deviation ( Q ), is given by sample standard deviation (s) as follows: IS 14277 : 1996 [ -cQ, x + tl-a(a-l)& I wheiezt - a/z denotes the value of standard normal deviate with the area of (1- F ) to its left and -$ is the standard error of X. NOTES 1 The value of zt_a/z can be obtained from the tables of the standard normal distribution given in IS 9300 (Part 2) : 1989. For95 percent and 99 percent confidence intervals, thevalue is 1.96 and 2.575, respectively. 2 As 1 - a increases, the length of the interval also increases. Also for a fixed a, as u increases the length of the interval increases. 3 For given a and a, larger the sample size, narrower is the length of the interval and better the estimate. Example 1 Measurements on the thickness (in codified units) of a sample of 16 mica discs from a production . process are as follows: 14,11,11,17,15,13,14,11,14,12,10,10,8,13,7,8 The mean of thickness values is calculated as 11.75. Suppose it is known that these discs have been produced by a controlled process whose standard deviation is known to be equal to 2.5. Following the procedure described in 5.1, we find that the 95 percent confidence interval for the mean is: 1.96 x 2.5 x 2.5 [ 11.75 _ 1.96 4 1, , .11.75 + 4 which gives us the interval [lo.& 13.01. The probability that the population mean is included in the estimated interval [lo.& 13.01 is 95 percent. Similarly, the 99 percent confidence interval can be calculated as [lO.l, 13.41. Example 2 Consider the above example when there is no knowledge about the standard deviation of the thickness. The estimate of ais obtained ass = 2.77. In this case the 95 percent confidence interval is seen to be [ 11*75 - t2*131) x $7 9 11-75 + t2-131) ' In some cases one-sided confidence intervals are most appropriate to use. The one-sided confidence interval may be either an upper limit or a lower limit. A 100 (l- a) percent one-sided confidence interval for population mean of a normally distributed variable with mean @) and variance (a) is given by: X - Zl-+; + QJ I [or X+zt-aU -*, K [ where ZI-~ denotes the value of standard normal 51.1 1 deviate with the area of (1 -a ) to its left and -$ is the standard error of ?. 5.2 Standard Deviation not Known When the parameter o is not known and if the sample is large,' say, greater than 30, its point estimate (s), as given in 4.2, is made use of in place ofa for calculating confidence interval as given in 5.1. If the sample size is less than 30, its point estimate (s), as given in 4.2 is made use of in place of 0, and student's &distribution is made use of in place of normal distribution for calculating confidence interval. The 100 (l- a) percent confidence interval is given by: [ which is [10.3,13.2]. Similarly, the 99 percent confidence interval can be, calculated as [9.7,13.8]. 6 CONFIDENCE INTERVAL FOR VARIANCE 6.1 The one-sided confidence interval for variance &t is defined by the upper limit, the lower limit being taken as equal to zero. me interval is'given by: x- tl-a/2(n-l)& ,F + ~I--a/2(n-I)& I Lo* *$$$I NOTE -The values of& _ a (" _t) are given in the tables for critical values of x2 distribution, for CIequal to 0.05 and 0.01 in IS 6200 (Part 2) . . : 1977. NOTE - The values of tt - M (n - 1) are given in the tables on critical values of r-distribution (two-sided) in IS 6200 (Part 1) : 1995. 5.2.1 For 100 (l- one-sided confidence interval a a) percent confidence interval is given by: + Q) ExumP1e 3 The precision of a micrometer is measured in terms of the standard deviation of the readings made by it. The 10 readings made on a specimen with a 2 x - t1-*(n-l)+ [- 1 IS 14277:1996 calibrated micrometer are as follows: 0.501, 0.502, 0.498, 0.499, 0.501, 0.503, 0.499, 0.502,0.497,0.504 We are required to find the 95 percent confidence interval for the variance, 2, of these readings. Thecalculatedvalueof(n-1)s2is0.0000464while thevalue ofX2 for n-l = 9 is 16.92 for confidence level of 95 percent. The one-sided 95 percent [0, 0.000 002 742). confidence interval is bt - t1 - a/2 (n-2) vs;(n blsy ' (n-2)s; 61 + Cl -a/2 (n-2) - blsx,t II sj? -2) s,2 where bl, sx and So are as defined in 4.3 and sy is the sample standard deviation ofy as defined for Y in 4.2. Example 4 The temperature in the Heating Zone of an Exhaust is related to the duration of heating. To estimate this relationship readings were taken on the temperature once every 3 minutes beginning 2 minutes after the heating started. These readings were as follows: 7 CONFIDENCE INTERVAL FOR THE REGRESSION COEFFICIENT 7.1 The confidence interval for bt is given by: Time (min) Temperature 2 110 5 130 8 160 11 180 14 190 17 210 20 220 , 23 250 26 260 29 280 We take x = duration in minutes, and Y = temperature in "C. The point-estimate of the regression coefficientpr, that is, bl is equal to 6.12. We also have sxy = 4 545 and sz = 742.5 and s;= 28 090. Thus, the 95 percent confidence interval is obtained as [ 6.12 - (2.306) x v s , 33.64 6.12 + (2.306) x v --I 742.5 = [ 6.12 - 2.306 x 0.213,6.12 + 2.306 x 0.213 ] This gives us the interval [5.6,6.6]. The regression coefficient in this example measures the rate of change in temperature per minute increase in duration of heating. There are 95 percent chances that this rate is between 5.6 and 6.6"C. IS 14277 : 1996 ANNEX A (Foreword> LIST OF INDIAN STANDARDS ON STATISTICAL METHODS 397 Title Method for statistical quality control during production : (Part 1) : 1972 Part 1 Control charts for variables Method for statistical quality 397 control during production : (Part 2) : 1985 Part 2 Control charts for attributes and count of defects Method for statistical quality 397 control during production : (Part 3) : 1980 Part 3 Special control charts Method for statistical quality 397 control during production : (Part 4) : 1987 Part 4 Master control systems Manual on basic principles of 1548 lot sampling : Part 1 Itemized (Part 1) : 1981 lot sampling Sampling inspection proce2500 dures : Part 1 Attribute sam(Part 1) : 1992/ IS0 2859-l : 1989 pling plans indexed by acceptable quality level (AQL) for lot-by-lot inspection Sampling inspection proce2500 dures : Part 2 Inspection by (Part 2) : 1965 variables for percent defective Sampling inspection proce2500 (Part 3) : 19951 dures : Part 3 Attribute samIS0 2859-2 : 1985 pling plans indexed by limiting quality (LQ) for isolated lot inspection Methods for random sampling 4905 : 1968 Guide on precision of test 5420 methods : Part 1 Principles (Part 1) : 1969 and applications Guide on precision of test 5420 methods : Part 2 Inter(Part 2) : 1973 laboratory testing Statistical tests of significance : 6200 Part 1 t-, Normal and F-tests (Part 1) : 1995 Statistical tests of significance : 6200 (Part 2) : 1977 Part 2 x2 -test Statistical tests of significance : 6200 Part 3 Tests for normality (Part 3) : 1984 Statistical tests of significance : 6200 Part 4 Non-parametric tests (Part 4) : 1983 IS No. IS No. 7200 (Part 1) : 1989 7200 (Part 2) : 1975 7300 : 1995 7600 : 1975 7920 (Part 1) : 1994 7920 (Part 2) : 1994 7920 (Part 3) : l995 8900 : 1978 9300 (Part 1) : 1979 9300 (Part 2) : 1989 10427 (Part 1) : 1982 10427 (Part 2) : 1986 10645 : 1983 12347 : 1988 12348 : 1988 13131: 1991 14277 : 1995 Title Presentation of statistical data : Part 1 Tabulation and summarization Presentation of statistical data : Part 2 Diagrammatic representation of data Methods for regression and correlation Analysis of variance Statistical vocabulary and symbols : Part 1 Probability and general statistical terms Statistical vocabulary and symbols : Part 2 Statistical quality control Statistical vocabulary and symbols : Part 3 Design of experiments (underprint) Criteria for the rejection of outlying observations models for inStatistical dustrial applications : Part 1 Discrete models Statistical models for industrial applications : Part 2 Continuous models Designs for industrial experimentation : Part 1 Standard designs Designs for industrial experimentation : Part 2 Orthogonal arrays Method for estimation of process capability Analysis of means-A graphical procedure Use of probability papers Statistical tolerance interval Methods for determination Statistical interpretation of test results - Estimation of mean, standard deviation and regression coefficient-Confidence interval statistical Handbook on qualit control SP 28 : 1994 IS 14277 : 1996 ANNEX B (Foreword) COMMITTEE COMPOSITION Statistical Methods for Quality and Reliability Sectional Committee, MSD 3 ChUk PROPS. P. MUKHERJEE University of Calcutta, Calcutta Msnbers SHRIS. K AGAR~AL SHRISURESH KUMAR (&?mu&) PROPM. L AGGARWAL PROFS. BLSWA~ (Altemate) DR A~HOK KUMAR SHRIM. G. Bwe SHRIA KIJMAR(Alremate) SHRIBHAGWAN D.&s SHRIM. BHO~MIK (Akmae) DR Representing Powergrid Corporation of India Limited, New Delhi University OC Delhi, Delhi Defence Science Centre, DRDO, Ministry of Defence, New Delhi Tata Iron and Steel Company Limited, Jamshedpur Central Statistical Organization, New Delhi Indian Association for Productivity, Quality and Reliability, Calcutta B. DAS DR DEBABRATARAY (Akmare) Tata Engineering and Locomotive Company Limited, Jamshedpur Quality Management Institute, New Delhi Indian Agricultural Statistics Research Institute, New Delhi Bharat Heavy Electricals Limited, Hyderabad Indian Jute Industries' Research Association, Calcutta Indian Statistical Institute, Calcutta Bajaj Auto Limited, Pune In personal capacity (B-109 Malviya Nagar, New D&ii Steel Authority of India Limited, New Delhi Lucas-TVS Limited, Madras Management and Statistics Centre, Madras Army Statistical Organization, New Delhi DCM Limited, New Delhi SRF Limited, Madras Director General, BIS (Er-uficio Member) MemberSecremy SHRIMATI BINDU MEHTA SHRIS. N. DAS SHRIG. W. DATEY DR 0. P. KAIHURIA DR PRAINESHU (Alternate) SHY A. ,V. KR~sHNAN SHRIJAVED AFAQUE (Akmafe) DR A LAHIRI SHRILJ. DUTTA(Alternate) PROFS. R. MOHAN PROPARVIND SEIH(Aftermu) SHRIM. M. NAIK SHRIS. P. DAM (Al&mate) PROFA. N. NANKANA SHRIK. K. PAHUJA SHRIGOPAL BANDHU (Alternate) SHRIC. RAHGANA~HAN SHRIC. v. SESHAGIRI (Al&male) AIR c&DE R. S.&4PAm (REID) SHRIS. C. SEDDEY SHIUA K SAHA(Akmufe) SHRID. R. SEN SHRI-1 SUBRAMAHIAM (Alternate) SHRIP. SUBBARAMAIAH SHRIA. K TALWAR, Director (MSD) Deputy Director (MSD), BIS (Continued on page 6) IS14277:1996 (Continued form page 5) Basic Statistical Methods Subcommittee, MSD 3 : 1 convener PROFS. P. MIJKI-ERJEE Members Repraenting University of Calcutta, Calcutta University of Delhi, Delhi Defence Science Centre, DRDO, Ministry of Defence, New Delhi Quality Management Institute, New Delhi Indian Jute Industries' Research Association, Calcutta Indian Statistical Institute, New Delhi Bajaj Auto Limited, Pune In personal capacity (B-I 09 Mahiya Nagar, New Delhi) Steel Authority of India Limited, New Delhi DCM Limited, New Delhi PROF M. L. AGGARWAL DR ASHOKKUMAR SH~U G. W. DATEY DRA.LAHIRI PROFS. R. MOHAN SHRIM. M. NA~K PROFA. N. NANKANA SHRtK. K. PAHIJJA SHtUD. R. SEN Panel for Basic Methods Including Terminology, MSD 3:1/P-l Convener SHR~ G. W. DATEY Members Quality Management Institute, New Delhi Defence Science Centre, DRDO, Ministry of Defence, New Delhi Indian Statistical Institute, New Delhi Indian Association for Productivity, Quality and Reliability, Calcutta DR ASHOKKUMAR DR S.R. MoHA~J DR D. RAY Ilureau of Indian Standards BIS is a statutory institution established under the Bureau oflndiun Standards Act, 1986 to promote harmonious development of the activities of standardization, marking and quality certification of goods and attending to connected matters in the country. Copyright BIS has the copyright of all its publications. No part of these publications may be reproduced in any form without the prior permission in writing of BIS. This does not preclude the free use, in the course of implementing the standard, of necessary details, such as symbols and sizes, type or grade designations. 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